There is the Oxyrhynchus Hymn, it is the oldest Christian Hymn to the Holy Trinity that we know of, that has the sheet notated, in Ancient Greek Vocal notation.
If non-church music is included too, then you have the Seikilos Epitaph, and other sheets from the Roman period. I recommend you the book from Pöhlmann & West (2001) "Documents of Ancient Greek music", look for the sheets from the Roman period, some are dated to be as late as 4th century AD.
Byzantine Music by itself doesn't exist.
We call our church music Ψαλτική Μουσική (Psaltic Music),
Our Secular Music depends on the local tradition, Greek music is just as Byzantine as is Turkish music, and so is the old music from the Balkans (if you are a Romanian compatriot, I recommend you listen to the Anton Pann ensemble); some would say that Greek and Turkish music are the same, but there are still differences that one can sometimes notice, notably in the instruments used, the interpretation style, the melismata etc.
And the Secular music, it is a continuation of the music of Roman times. You can see that Al-Farabi used an alphabetic notation that is almost the same as the Ancient Greek vocal notation, and it definitely a descendant of the Ancient Greek musical notation:
Marcel Camprubí Arabic music theory from the Abbasid period (750–1258) is a rich and sophisticated tradition that incorporates ancient Greek thought into autochthonous Arabic musical theorization a…
Techniques change over time, so does composition style, but it is still the same type of music.
So, you don't have Ancient Greek music from the 4th century AD vs Byzantine music from the 4th century AD, as two different musical styles, if you went back in time, you would just see a late evolution of Ancient Greek Music.
Therefore, the more you go into the past, the more it should resemble Ancient Greek compositions from the Hellenistic era, and the less it will resemble the music of the Modern Middle East.